Aircraft: CESSNA 150
Phase of flight: Initial Approach
Anomaly: Airspace Violation All Types; Conflict Airborne Conflict; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Other / Unknown

Non-towered airport traffic pattern. Aircraft X is on downwind. Aircraft Y is on the 45; tucks in nicely behind Aircraft X on downwind with proper separation and radio calls. Aircraft X flies an exceptionally long downwind; nearly 2 NM past the runway threshold; for no apparent reason (no one ahead of them; no one waiting for takeoff). Radio calls from Aircraft X are unintelligible due to strong accent; neither pilot in Aircraft Y can understand them other than the name of the airport. Seeing Aircraft X so far out on downwind; not being sure if they were departing or just flying a huge pattern; Aircraft Y turns base at the normal time (roughly 0.5 NM past the runway threshold) and calls it out on CTAF. Appears to be plenty of spacing. Aircraft Y is not intending to 'cut off' Aircraft X. Plenty of time for Aircraft Y to land; touch-n-go; or go-around without impeding Aircraft X's plans at all. Remember; similar types; therefore similar speeds. With as little as 1 NM separation; that's a full minute in trail at a reasonable 150/152 approach speed of 60 kts; slightly less than a minute at let's say 70 kts. Nonetheless; someone might interpret Aircraft Y's action as taking 'advantage of this rule to cut in front of another which is on final approach to land or to overtake that aircraft;' (FAR 91.113) even though Aircraft X was not on final approach; and we (Aircraft Y) clearly were not 'overtaking' them. As soon as we started turning base; we noticed visually (since we could not understand their transmissions) that they turned base at roughly the same time. Here's where it gets crazy. We saw on our ADS-B In display that they were gaining on us while we were both on final approach. Not wanting to hold them up at all; we executed a go-around from roughly 200 feet AGL and announced our go-around on the CTAF. We then lost track of Aircraft X temporarily while focusing on the go-around procedure. Once we re-established ourselves on downwind; we noticed that Aircraft Y had come up behind us and was GAINING on us on downwind. Without any radio calls from Aircraft X; to boot! We in Aircraft Y were making the usual radio calls 'turning crosswind;' 'turning downwind;' 'on downwind.' Nothing but silence from Aircraft X. Our ADS-B In told us they were same altitude and ZERO NM behind us. My student looked back and did catch a glimpse of them; not sure how far behind. On flightradar24; it appears Aircraft X caught up with Aircraft Y to within 1/2 NM; although I suspect it was even closer. With both of us confused and getting nervous; my student elected to depart the pattern on effectively a downwind departure; and I agreed. I instructed full power; carb heat off; flaps up to give us a chance of outrunning the closing Aircraft X aircraft behind us. We announced our departure on CTAF and turned slightly right in case Aircraft X simply did not see us (fat chance). That way; we would be out of their way if they continued straight on the downwind. Our tactic did manage to provide some extra distance from Aircraft X; but get this - Aircraft X turned slightly right just like we did and appeared to be chasing us! Again; flightradar24 shows these maneuvers clearly. After a few seconds; Aircraft X appeared to give up the chase and returned via a slight left turn back to the downwind while we climbed out on the heading discussed above (approximately 20 degrees east of the downwind heading). Again; we called our position and intentions on the CTAF stating that there was a traffic conflict we were unable to resolve; so we were exiting the pattern to the <direction>. In my opinion as pilot in command; the closing-from-behind aircraft presented a clear and present danger. While I did not advise ATC (nor were we in contact with any ATC facility to declare to); I exercised my authority per FAR 91.3(b) to the extent required to meet the situation. Thus; the non-standard exit from the traffic pattern; in other words not one of the exits recommended by AC 90-66B(or whatever suffix it' s up to nowadays). Flight Instructor reported a NORAD aircraft in pattern traffic overtaking them resulted in pilot exiting traffic pattern to avoid conflict.